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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 195–206 | Cite as

Presentation of new thermal conductivity expression for \(\hbox {Al}_2\hbox {O}_3\)–water and \(\hbox {CuO}\)–water nanofluids using gene expression programming (GEP)

  • Saber Yekani Motlagh
  • Abbas SharifiEmail author
  • Mohsen Ahmadi
  • Homayoun Badfar
Article

Abstract

In the present investigation, gene expression programming (GEP) is used to predict thermal conductivity of nanofluids consisting of \(\hbox {Al}_2\hbox {O}_3\) and CuO nanoparticles suspended in water. The obtained new model is a function of temperature, volume fraction, and diameter of the nanoparticles. To predict the thermal conductivity, experimental data from literatures were partitioned into two sets: a train (800 numbers) and a test (200 numbers) data sets. The model was designed by the train set, and the results were compared with the test data set. The estimated heat conductivity was compared with experimental data and several most cited relations in the literature. Moreover, the new thermal conductivity model was tested in the simulation of benchmark case study of nanofluid free convection inside the square cavity. The predicted Nusselt number was compared with available experimental data at five Rayleigh numbers. The findings illustrated that the GEP can estimate and model the thermal conductivity of nanofluid very efficiently, and it can be used successfully for simulation of engineering problems.

Graphical Abstract

Keywords

Nanofluid Thermal conductivity Genetic expression programming Modeling 

Statistic

AIC

The Akaike information criterion

ANN

Artificial neural network

CC

Correlation coefficient

GEP

Gene expression programming

MSE

Mean square error \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_\mathrm{j})^2/n\)

RMSE

Root-mean-square error \(\sqrt{\hbox {MSE}}\)

RNC

Random numerical constant

\(R^2\)

Coefficient of determination \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_{\mathrm{avg}})/\sum _{j=1}^{n}(T_{\mathrm{j}}-T_{\mathrm{avg}})\)

SSE

Sum of squared error \(\sum _{j=1}^{n}(C_{\mathrm{j}}-T_\mathrm{j})^2\)

Genetic

a, b, c

Terminals as temperature, volume fraction, and diameter, respectively

\(C_{\mathrm{ij}}\)

The value predicted by chromosome i for fitness case j

\(f_\mathrm{i}\)

Fitness function

\(G_\mathrm{i}C_\mathrm{j}\)

The jth constant of ith gene

LCPGQE

Natural logarithm, cosine, power, logarithm, square root, exponential functions

\(T_\mathrm{j}\)

The target value

?

Random numerical constants

Fluid mechanic

\(C_{\mathrm{p}}\)

Specific heat at constant pressure

d

Particle diameter

g

Gravity

H

Characteristic length

l

Mean free path

k

Thermal conductivity

\(k_\mathrm{B}\)

Boltzmann constant \(=1.38064852\times 10^{-23}\) m\(^2\) kg s−2 K−1

Nu

Nusselt number (hL)/k

Pr

Prandtl number \((C_{\mathrm{p}}\mu )/ k\)

Ra

Rayleigh number \((g \beta Pr \Delta T L^2)/\nu ^2\)

\(Re_\mathrm{B}\)

Brownian Reynolds number \(\frac{\rho _{\mathrm{bf}}k_\mathrm{B} T}{3 \pi \mu ^2 l_{\mathrm{bf}}}\)

T

Fluid temperature

V

Fluid velocity

Indexes

Ave, \(h,c,*,r\)

Average, hot, cold, non-dimensional, relative

fnfbfp

Fluid, nanofluid, base fluid, particle

Greece symbol

\(\alpha\)

Thermal diffusion coefficient

\(\beta\)

Thermal expansion coefficient

\(\psi\)

Sphericity

\(\phi\)

Volume fraction

\(\sigma ^2\)

Residual variance

\(\mu\)

Kinematic viscosity

\(\nu\)

Dynamic viscosity

\(\rho\)

Fluid density

References

  1. 1.
    Masuda H, Ebata A, Teramae K, Hishinuma N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei. 1993;7(4):227.Google Scholar
  2. 2.
    Lee J, Lee H, Baik YJ, Koo J. Quantitative analyses of factors affecting thermal conductivity of nanofluids using an improved transient hot-wire method apparatus. Int J Heat Mass Transf. 2015;89:116.Google Scholar
  3. 3.
    Shaker M, Birgersson E, Mujumdar A. Extended Maxwell model for the thermal conductivity of nanofluids that accounts for nonlocal heat transfer. Int J Therm Sci. 2014;84:260.Google Scholar
  4. 4.
    Zerradi H, Ouaskit S, Dezairi A, Loulijat H, Mizani S. New Nusselt number correlations to predict the thermal conductivity of nanofluids. Adv Powder Technol. 2014;25(3):1124.Google Scholar
  5. 5.
    Esfe MH, Saedodin S, Akbari M, Karimipour A, Afrand M, Wongwises S, Safaei MR, Dahari M. Experimental investigation and development of new correlations for thermal conductivity of CuO/EG-water nanofluid. Int Commun Heat Mass Transf. 2015;65:47.Google Scholar
  6. 6.
    Esfe MH, Afrand M, Yan WM, Akbari M. Applicability of artificial neural network and nonlinear regression to predict thermal conductivity modeling of Al\(_2\)O\(_3\) water nanofluids using experimental data. Int Commun Heat Mass Transf. 2015;66:246.Google Scholar
  7. 7.
    Esfe MH, Wongwises S, Naderi A, Asadi A, Safaei MR, Rostamian H, Dahari M, Karimipour A. Thermal conductivity of Cu/TiO\(_2\)-water/EG hybrid nanofluid: experimental data and modeling using artificial neural network and correlation. Int Commun Heat Mass Transf. 2015;66:100.Google Scholar
  8. 8.
    Ariana M, Vaferi B, Karimi G. Prediction of thermal conductivity of alumina water-based nanofluids by artificial neural networks. Powder Technol. 2015;278:1.Google Scholar
  9. 9.
    Mechiri SK, Vasu V, Babu S. Thermal conductivity of Cu–Zn hybrid Newtonian nanofluids: experimental data and modeling using neural network. Proc Eng. 2015;127:561.Google Scholar
  10. 10.
    Mehrabi M, Sharifpur M, Meyer J. Application of the FCM-based neuro-fuzzy inference system and genetic algorithm-polynomial neural network approaches to modelling the thermal conductivity of aluminawater nanofluids. Int Commun Heat Mass Transf. 2012;39(7):971.Google Scholar
  11. 11.
    Longo GA, Zilio C, Ceseracciu E, Reggiani M. Application of artificial neural network (ANN) for the prediction of thermal conductivity of oxide–water nanofluids. Nano Energy. 2012;1(2):290.Google Scholar
  12. 12.
    Esfe MH, Saedodin S, Sina N, Afrand M, Rostami S. Designing an artificial neural network to predict thermal conductivity and dynamic viscosity of ferromagnetic nanofluid. Int Commun Heat Mass Transf. 2015;68:50.Google Scholar
  13. 13.
    Zadkhast M, Toghraie D, Karimipour A. Developing a new correlation to estimate the thermal conductivity of MWCNT-CuO/water hybrid nanofluid via an experimental investigation. J Therm Anal Calorim. 2017;129(2):859.  https://doi.org/10.1007/s10973-017-6213-8.Google Scholar
  14. 14.
    Hemmat Esfe M, Saedodin S, Wongwises S, Toghraie D. An experimental study on the effect of diameter on thermal conductivity and dynamic viscosity of Fe/water nanofluids. J Therm Anal Calorim. 2015;119(3):1817.  https://doi.org/10.1007/s10973-014-4328-8.Google Scholar
  15. 15.
    Hemmat Esfe M, Esfandeh S, Rejvani M. Modeling of thermal conductivity of MWCNT–SiO\(_2\) (30:70%)/EG hybrid nanofluid, sensitivity analyzing and cost performance for industrial applications. J Therm Anal Calorim. 2017;131(2):1437.  https://doi.org/10.1007/s10973-017-6680-y.Google Scholar
  16. 16.
    Hemmat Esfe M, Saedodin S, Bahiraei M, Toghraie D, Mahian O, Wongwises S. Thermal conductivity modeling of MgO/EG nanofluids using experimental data and artificial neural network. J Therm Anal Calorim. 2014;118(1):287.  https://doi.org/10.1007/s10973-014-4002-1.Google Scholar
  17. 17.
    Hemmat Esfe M, Rostamian H, Toghraie D, Yan WM. Using artificial neural network to predict thermal conductivity of ethylene glycol with alumina nanoparticle: effects of temperature and solid volume fraction. J Therm Anal Calorim. 2016;126(2):643.  https://doi.org/10.1007/s10973-016-5506-7.Google Scholar
  18. 18.
    Hemmat Esfe M, Ahangar MRH, Toghraie D, Hajmohammad MH, Rostamian H, Tourang H. Designing artificial neural network on thermal conductivity of Al\(_2\)O\(_3\)–water–EG (60–40%) nanofluid using experimental data. J Therm Anal Calorim. 2016;126(2):837.  https://doi.org/10.1007/s10973-016-5469-8.Google Scholar
  19. 19.
    Toghraie D, Chaharsoghi VA, Afrand M. Measurement of thermal conductivity of ZnO-TiO\(_2\)/EG hybrid nanofluid. J Therm Anal Calorim. 2016;125(1):527.  https://doi.org/10.1007/s10973-016-5436-4.Google Scholar
  20. 20.
    Akbari OA, Afrouzi HH, Marzban A, Toghraie D, Malekzade H, Arabpour A. Investigation of volume fraction of nanoparticles effect and aspect ratio of the twisted tape in the tube. J Therm Anal Calorim. 2017;129(3):1911.  https://doi.org/10.1007/s10973-017-6372-7.Google Scholar
  21. 21.
    Hamilton RL, Crosser OK. Thermal conductivity of hetrogeneous two-component systems. Ind Eng Chem Fundam. 1962;1(3):187.Google Scholar
  22. 22.
    Koo J, Kleinstreuer C. A new thermal conductivity model for nanofluids. J Nanopart Res. 2005;6(6):577.Google Scholar
  23. 23.
    Maxwell JC. Electricity and magnetism, vol. 1. Clarendon press; 1954.Google Scholar
  24. 24.
    Das SK, Putra N, Thiesen P, Roetzel W. Temperature dependence of thermal conductivity enhancement for nanofluids. J Heat Transf. 2003;125(4):567.Google Scholar
  25. 25.
    Chon CH, Kihm KD, Lee SP, Choi SUS. Empirical correlation finding the role of temperature and particle size for nanofluid (Al\(_2\)O\(_3\)) thermal conductivity enhancement. Appl Phys Lett. 2005;87(15):153107.Google Scholar
  26. 26.
    Li CH, Peterson GP. Experimental investigation of temperature and volume fraction variations on the effective thermal conductivity of nanoparticle suspensions (nanofluids). J Appl Phys. 2006;99(8):084314.Google Scholar
  27. 27.
    Mintsa HA, Roy G, Nguyen CT, Doucet D. New temperature dependent thermal conductivity data for water-based nanofluids. Int J Therm Sci. 2009;48(2):363.Google Scholar
  28. 28.
    Cramer NL. A representation for the adaptive generation of simple sequential programs. In: Proceedings of the 1st international conference on genetic algorithms, L. Erlbaum Associates Inc., Hillsdale, NJ, USA; 1985, p. 183–7.Google Scholar
  29. 29.
    Koza JR. Genetic programming: on the programming of computers by means of natural selection, vol. 1. Cambridge: MIT Press; 1992.Google Scholar
  30. 30.
    Stefanini T. The genetic coding of behavioral attributes in cellular automata. In: Artificial Life at Stanford 1994, Stanford Bookstore, Stanford, California, 94305-3079 USA; 1994, p. 172–80.Google Scholar
  31. 31.
    Ferreira C. Soft computing and industry: recent applications. London: Springer; 2002 (chap. Gene Expression Programming in Problem Solving, pp. 635–653).Google Scholar
  32. 32.
    Akaike H. Selected papers of hirotugu akaike. New York: Springer; 1998 (chap. Information Theory and an Extension of the Maximum Likelihood Principle, pp. 199–213).Google Scholar
  33. 33.
    Ho C, Liu W, Chang Y, Lin C. Natural convection heat transfer of alumina–water nanofluid in vertical square enclosures: an experimental study. Int J Therm Sci. 2010;49(8):1345.Google Scholar
  34. 34.
    Corcione M. Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers Manag. 2011;52(1):789.Google Scholar
  35. 35.
    Corcione M. Heat transfer features of buoyancy-driven nanofluids inside rectangular enclosures differentially heated at the sidewalls. Int J Therm Sci. 2010;49(9):1536.Google Scholar
  36. 36.
    Garoosi F, Garoosi S, Hooman K. Numerical simulation of natural convection and mixed convection of the nanofluid in a square cavity using Buongiorno model. Powder Technol. 2014;268:279.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUrmia University of Technology (UUT)UrmiaIran
  2. 2.Department of Industrial EngineeringUrmia University of Technology (UUT)UrmiaIran

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